There are several generalizations of the classical theory of Sobolev spaces as they are necessary for the applications to Carnot-Carathéodory spaces, subelliptic equations, quasiconformal mappings on Carnot groups and more general Loewner spaces, analysis on topological manifolds, potential theory on infinite graphs, analysis on fractals and the theory of Dirichlet forms. The aim of this paper is to present a unified approach to the theory of Sobolev spaces that covers applications to many of those areas. The variety of different areas of applications forces a very general setting. We are given a metric space X equipped with a doubling measure ¯. A generalization of a Sobolev function and its gradient is a pair u 2 L 1 loc (X), 0 g 2 L p (X) such that for every ball B ae X the Poincar'e-type inequality Z B ju \Gamma uB j d¯ Cr `Z oeB g p d¯ ' 1=p holds, where r is the radius of B and oe 1, C ? 0 are fixed constants. Working in the above setting we show that basically...

preliminaries Let H be a complex Hilbert space, A a densely dened linear operator in H (the domain of A will be denoted D(A)). Suppose that A has a closure A or, equivalently, that the adjoint operator A is densely dened (see e.g. [32]). We shall denote by GA the graph of A i.e. the set of pairs fu; Aug; u 2 D(A). Then G A = GA , i.e. the graph of A is the closure of the graph of A. Moreover A = A = (A ) . Now let A + be another densely dened linear operator in H. DEFINITION 1.1. A + is called formally adjoint to A if (1:1) (Au; v) = (u; A + v); u 2 D(A); v 2 D(A + ); where (; ) is the scalar product in H. If A = A + then A is called symmetric or formally self{adjoint. Note that since A; A + are densely dened, both A and A + have closures. DEFINITION 1.2. Let A; A + be as in Denition 1.1. Then the minimal and the maximal operator for A are dened as follows: A min = A = A ; A max = (A + ) : Note that both A min and A max are...

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We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

Abstract. We obtain a compactness result for various classes of Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual metrics, Kähler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding metrics with orbifold singularities. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs substantially from the Einstein case in that we do not assume any pointwise Ricci curvature bound. 1.

We give upper estimates on the long time behaviour of the heat kernel on a non-compact Riemannian manifold and infinite graphs, which only depend on a lower bound of the volume growth. We also show that these estimates are optimal.

Abstract. This is a survey on the convergence theory developed rst by Cheeger and Gromov. In their theory one is concerned with the compactness of the class of riemannian manifolds with bounded curvature and lower bound on the injectivity radius. We explain and give proofs of almost all the major results, including Anderson's generalizations to the case where all one has is bounded Ricci curvature. The exposition is streamlined by the introduction of a norm for riemannian manifolds, which makes the theory more like that of Holder and Sobolev spaces. 1.

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80’s, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.

Abstract. We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric. 1.

Abstract. For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the Laplace-Beltrami operator on Riemannian manifolds for p> 2. 1.